Homotopies and the Universal Fixed Point Property

TY - JOUR

T1 - Homotopies and the Universal Fixed Point Property

AU - Szymik, Markus

PY - 2015/11/1

Y1 - 2015/11/1

N2 - A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular to the analysis of convex subspaces of Banach spaces, to the topology of finite-dimensional manifolds and CW complexes, and to the combinatorics of Kolmogorov spaces associated with finite posets.

AB - A topological space has the fixed point property if every continuous self-map of that space has at least one fixed point. We demonstrate that there are serious restraints imposed by the requirement that there be a choice of fixed points that is continuous whenever the self-map varies continuously. To even specify the problem, we introduce the universal fixed point property. Our results apply in particular to the analysis of convex subspaces of Banach spaces, to the topology of finite-dimensional manifolds and CW complexes, and to the combinatorics of Kolmogorov spaces associated with finite posets.

KW - Finite poset

KW - Fixed point property

KW - Homotopy

KW - Kolmogorov space

UR - http://www.scopus.com/inward/record.url?scp=84944351337&partnerID=8YFLogxK

U2 - 10.1007/s11083-014-9332-x

DO - 10.1007/s11083-014-9332-x

M3 - Journal article

AN - SCOPUS:84944351337

SN - 0167-8094

VL - 32

SP - 301

EP - 311

JO - Order

JF - Order

IS - 3

ER -